Cycle graphThe four hexagonal cycles have the inversion (the black knot on top) in common. The hexagons are symmetric, so e.g. 3 and 4 are in the same cycle, but not 3 and 12.

A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual of an octahedron.

The group of orientation-preserving symmetries is S4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four pairs of opposite faces of the octahedron.

As the hyperoctahedral group of dimension 3 the full octahedral group is the wreath productS2≀S3≃S23⋊S3{\displaystyle S_{2}\wr S_{3}\simeq S_{2}^{3}\rtimes S_{3}},and a natural way to identify its elements is as pairs (m,n){\displaystyle (m,n)} with m∈[0,23){\displaystyle m\in [0,2^{3})} and n∈[0,3!){\displaystyle n\in [0,3!)}.But as it is also the direct productS4×S2{\displaystyle S_{4}\times S_{2}}, one can simply identify the elements of tetrahedral subgrup Td as a∈[0,4!){\displaystyle a\in [0,4!)} and their inversions as a′{\displaystyle a'}.

So e.g. the identity (0,0){\displaystyle (0,0)} is represented as 0{\displaystyle 0} and the inversion (7,0){\displaystyle (7,0)} as 0′{\displaystyle 0'}.(3,1){\displaystyle (3,1)} is represented as 6{\displaystyle 6} and (4,1){\displaystyle (4,1)} as 6′{\displaystyle 6'}.

O, 432, or [4,3]+ of order 24, is chiral octahedral symmetry or rotational octahedral symmetry . This group is like chiral tetrahedral symmetryT, but the C2 axes are now C4 axes, and additionally there are 6 C2 axes, through the midpoints of the edges of the cube. Td and O are isomorphic as abstract groups: they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion. O is the rotation group of the cube and the regular octahedron.

Oh, *432, [4,3], or m3m of order 48 - achiral octahedral symmetry or full octahedral symmetry. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group is isomorphic to S4.C4, and is the full symmetry group of the cube and octahedron. It is the hyperoctahedral group for n = 3. See also the isometries of the cube.

With the 4-fold axes as coordinate axes, a fundamental domain of Oh is given by 0 ≤ x ≤ y ≤ z. An object with this symmetry is characterized by the part of the object in the fundamental domain, for example the cube is given by z = 1, and the octahedron by x + y + z = 1 (or the corresponding inequalities, to get the solid instead of the surface).
ax + by + cz = 1 gives a polyhedron with 48 faces, e.g. the disdyakis dodecahedron.

Faces are 8-by-8 combined to larger faces for a = b = 0 (cube) and 6-by-6 for a = b = c (octahedron).

The 9 mirror lines of full octahedral symmetry can be divided into two subgroups of 3 and 6 (drawn in purple and red), representing in two orthogonal subsymmetries: D2h, and Td. D2h symmetry can be doubled to D4h by restoring 2 mirrors from one of three orientations.

Take the set of all 3x3 permutation matrices and assign a + sign or a - sign to each of the three 1s. There are 6 matrices x 8 sign permutations = 48 matrices in total giving the full octahedral group. There are exactly 24 matrices with determinant = +1 and these are the rotation matrices of the chiral octahedral group. The other 24 matrices correspond to a reflection or inversion.

Three reflectional generator matices are needed for octahedral symmetry, which represent the three mirrors of a Coxeter-Dynkin diagram. The product of the reflections produce 3 rotational generators.

rotation about an axis from the center of an edge to the center of the opposite edge by an angle of 180°: 6 axes, 1 per axis, together 6 ((1 2), etc.; ((i±j)/√2, etc.)

rotation about a body diagonal by an angle of 120°: 4 axes, 2 per axis, together 8 ((1 2 3), etc.; (1±i±j±k)/2)

The same with inversion (x is mapped to −x) (also 24 isometries). Note that rotation by an angle of 180° about an axis combined with inversion is just reflection in the perpendicular plane. The combination of inversion and rotation about a body diagonal by an angle of 120° is rotation about the body diagonal by an angle of 60°, combined with reflection in the perpendicular plane (the rotation itself does not map the cube to itself; the intersection of the reflection plane with the cube is a regular hexagon).

An isometry of the cube can be identified in various ways:

by the faces three given adjacent faces (say 1, 2, and 3 on a die) are mapped to

by the image of a cube with on one face a non-symmetric marking: the face with the marking, whether it is normal or a mirror image, and the orientation

by a permutation of the four body diagonals (each of the 24 permutations is possible), combined with a toggle for inversion of the cube, or not

For cubes with colors or markings (like dice have), the symmetry group is a subgroup of Oh.

Examples:

C4v, [4], (*422): if one face has a different color (or two opposite faces have colors different from each other and from the other four), the cube has 8 isometries, like a square has in 2D.

D2h, [2,2], (*222): if opposite faces have the same colors, different for each set of two, the cube has 8 isometries, like a cuboid.

D4h, [4,2], (*422): if two opposite faces have the same color, and all other faces have one different color, the cube has 16 isometries, like a square prism (square box).

C2v, [2], (*22):

if two adjacent faces have the same color, and all other faces have one different color, the cube has 4 isometries.

if three faces, of which two opposite to each other, have one color and the other three one other color, the cube has 4 isometries.

if two opposite faces have the same color, and two other opposite faces also, and the last two have different colors, the cube has 4 isometries, like a piece of blank paper with a shape with a mirror symmetry.

Cs, [ ], (*):

if two adjacent faces have colors different from each other, and the other four have a third color, the cube has 2 isometries.

if two opposite faces have the same color, and all other faces have different colors, the cube has 2 isometries, like an asymmetric piece of blank paper.

C3v, [3], (*33): if three faces, of which none opposite to each other, have one color and the other three one other color, the cube has 6 isometries.

For some larger subgroups a cube with that group as symmetry group is not possible with just coloring whole faces. One has to draw some pattern on the faces.

Examples:

D2d, [2+,4], (2*2): if one face has a line segment dividing the face into two equal rectangles, and the opposite has the same in perpendicular direction, the cube has 8 isometries; there is a symmetry plane and 2-fold rotational symmetry with an axis at an angle of 45° to that plane, and, as a result, there is also another symmetry plane perpendicular to the first, and another axis of 2-fold rotational symmetry perpendicular to the first.

Th, [3+,4], (3*2): if each face has a line segment dividing the face into two equal rectangles, such that the line segments of adjacent faces do not meet at the edge, the cube has 24 isometries: the even permutations of the body diagonals and the same combined with inversion (x is mapped to −x).

Td, [3,3], (*332): if the cube consists of eight smaller cubes, four white and four black, put together alternatingly in all three standard directions, the cube has again 24 isometries: this time the even permutations of the body diagonals and the inverses of the other proper rotations.

T, [3,3]+, (332): if each face has the same pattern with 2-fold rotational symmetry, say the letter S, such that at all edges a top of one S meets a side of the other S, the cube has 12 isometries: the even permutations of the body diagonals.

The full symmetry of the cube, Oh, [4,3], (*432), is preserved if and only if all faces have the same pattern such that the full symmetry of the square is preserved, with for the square a symmetry group, Dih4, [4], of order 8.

The full symmetry of the cube under proper rotations, O, [4,3]+, (432), is preserved if and only if all faces have the same pattern with 4-fold rotational symmetry, C4, [4]+.

In Riemann surface theory, the Bolza surface, sometimes called the Bolza curve, is obtained as the ramified double cover of the Riemann sphere, with ramification locus at the set of vertices of the regular inscribed octahedron. Its automorphism group includes the hyperelliptic involution which flips the two sheets of the cover. The quotient by the order 2 subgroup generated by the hyperelliptic involution yields precisely the group of symmetries of the octahedron. Among the many remarkable properties of the Bolza surface is the fact that it maximizes the systole among all genus 2 hyperbolic surfaces.

1.
Dihedral symmetry in three dimensions
–
In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn. There are 3 types of symmetry in three dimensions, each shown below in 3 notation, Schönflies notation, Coxeter notation. For n = ∞ they correspond to three frieze groups, Schönflies notation is used, with Coxeter notation in brackets, and orbifold notation in parentheses. The term horizontal is used with respect to an axis of rotation. In 2D the symmetry group Dn includes reflections in lines, in 3D the two operations are distinguished, the group Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order, with reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis we have Dnh. Dnd, has vertical mirror planes between the rotation axes, not through them. As a result the vertical axis is a 2n-fold rotoreflection axis, Dnh is the symmetry group for a regular n-sided prisms and also for a regular n-sided bipyramid. Dnd is the group for a regular n-sided antiprism. Dn is the group of a partially rotated prism. D2 +, of order 4 is one of the three symmetry group types with the Klein four-group as abstract group and it has three perpendicular 2-fold rotation axes. It is the group of a cuboid with an S written on two opposite faces, in the same orientation. D2h, of order 8 is the group of a cuboid D2d. For Dnh, order 4n Cnh, order 2n Cnv, order 2n Dn, +, order 2n For Dnd, order 4n S2n, order 2n Cnv, order 2n Dn, +, cS1 maint, Multiple names, authors list N. W. Johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups Conway, John Horton, Huson, Daniel H

2.
Tetrahedral symmetry
–
A regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation. The set of orientation-preserving symmetries forms a group referred to as the alternating subgroup A4 of S4, chiral and full are discrete point symmetries. They are among the point groups of the cubic crystal system. Seen in stereographic projection the edges of the tetrakis hexahedron form 6 circles in the plane, each of these 6 circles represent a mirror line in tetrahedral symmetry. The intersection of these meet at order 2 and 3 gyration points. T,332, +, or 23, of order 12 – chiral or rotational tetrahedral symmetry, there are three orthogonal 2-fold rotation axes, like chiral dihedral symmetry D2 or 222, with in addition four 3-fold axes, centered between the three orthogonal directions. This group is isomorphic to A4, the group on 4 elements, in fact it is the group of even permutations of the four 3-fold axes. The three elements of the latter are the identity, clockwise rotation, and anti-clockwise rotation, corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. Td, *332, or 43m, of order 24 – achiral or full tetrahedral symmetry and this group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S4 axes, td and O are isomorphic as abstract groups, they both correspond to S4, the symmetric group on 4 objects. Td is the union of T and the set obtained by combining each element of O \ T with inversion, see also the isometries of the regular tetrahedron. This group has the same axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S6 axes, and there is an inversion symmetry. Th is isomorphic to T × Z2, every element of Th is either an element of T, apart from these two normal subgroups, there is also a normal subgroup D2h, of type Dih2 × Z2 = Z2 × Z2 × Z2. It is the product of the normal subgroup of T with Ci. The quotient group is the same as above, of type Z3, the three elements of the latter are the identity, clockwise rotation, and anti-clockwise rotation, corresponding to permutations of the three orthogonal 2-fold axes, preserving orientation. It is the symmetry of a cube with on each face a line segment dividing the face into two rectangles, such that the line segments of adjacent faces do not meet at the edge. The symmetries correspond to the permutations of the body diagonals

3.
Icosahedral symmetry
–
A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the set of symmetries, since it is the dual of the icosahedron. The set of orientation-preserving symmetries forms a group referred to as A5, the latter group is also known as the Coxeter group H3, and is also represented by Coxeter notation, and Coxeter diagram. Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups. Presentations corresponding to the above are, I, ⟨ s, t ∣ s 2, t 3,5 ⟩ I h, ⟨ s, t ∣ s 3 −2, t 5 −2 ⟩ and these correspond to the icosahedral groups being the triangle groups. The first presentation was given by William Rowan Hamilton in 1856, note that other presentations are possible, for instance as an alternating group. The icosahedral rotation group I is of order 60, the group I is isomorphic to A5, the alternating group of even permutations of five objects. This isomorphism can be realized by I acting on various compounds, notably the compound of five cubes, the group contains 5 versions of Th with 20 versions of D3, and 6 versions of D5. The full icosahedral group Ih has order 120 and it has I as normal subgroup of index 2. The group Ih is isomorphic to I × Z2, or A5 × Z2, with the inversion in the corresponding to element. Ih acts on the compound of five cubes and the compound of five octahedra and it acts on the compound of ten tetrahedra, I acts on the two chiral halves, and −1 interchanges the two halves. Notably, it does not act as S5, and these groups are not isomorphic, the group contains 10 versions of D3d and 6 versions of D5d. I is also isomorphic to PSL2, but Ih is not isomorphic to SL2, all of these classes of subgroups are conjugate, and admit geometric interpretations. Note that the stabilizer of a vertex/edge/face/polyhedron and its opposite are equal, stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate. Stabilizers of a pair of edges in Ih give Z2 × Z2 × Z2, there are 5 of these, stabilizers of an opposite pair of faces can be interpreted as stabilizers of the anti-prism they generate. g. Flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, in aluminum, the icosahedral structure was discovered experimentally three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011. Icosahedral symmetry is equivalently the projective linear group PSL, and is the symmetry group of the modular curve X. The modular curve X is geometrically a dodecahedron with a cusp at the center of each polygonal face, similar geometries occur for PSL and more general groups for other modular curves

4.
Octahedron
–
In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. A regular octahedron is a Platonic solid composed of eight equilateral triangles, a regular octahedron is the dual polyhedron of a cube. It is a square bipyramid in any of three orthogonal orientations and it is also a triangular antiprism in any of four orientations. An octahedron is the case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan metric, the second and third correspond to the B2 and A2 Coxeter planes. The octahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. An octahedron with edge length √2 can be placed with its center at the origin and its vertices on the coordinate axes, the Cartesian coordinates of the vertices are then. In an x–y–z Cartesian coordinate system, the octahedron with center coordinates, additionally the inertia tensor of the stretched octahedron is I =. These reduce to the equations for the regular octahedron when x m = y m = z m = a 22, the interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, an octahedron is the result of cutting off from a regular tetrahedron. One can also divide the edges of an octahedron in the ratio of the mean to define the vertices of an icosahedron. There are five octahedra that define any given icosahedron in this fashion, octahedra and tetrahedra can be alternated to form a vertex, edge, and face-uniform tessellation of space, called the octet truss by Buckminster Fuller. This is the only such tiling save the regular tessellation of cubes, another is a tessellation of octahedra and cuboctahedra. The octahedron is unique among the Platonic solids in having a number of faces meeting at each vertex. Consequently, it is the member of that group to possess mirror planes that do not pass through any of the faces. Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid, truncation of two opposite vertices results in a square bifrustum. The octahedron is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices and it is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size

5.
Cube
–
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids and it has 6 faces,12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and it is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron and it has cubical or octahedral symmetry. The cube has four special orthogonal projections, centered, on a vertex, edges, face, the first and third correspond to the A2 and B2 Coxeter planes. The cube can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. In analytic geometry, a surface with center and edge length of 2a is the locus of all points such that max = a. For a cube of length a, As the volume of a cube is the third power of its sides a × a × a, third powers are called cubes, by analogy with squares. A cube has the largest volume among cuboids with a surface area. Also, a cube has the largest volume among cuboids with the same linear size. They were unable to solve this problem, and in 1837 Pierre Wantzel proved it to be impossible because the root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the faces around each vertex,111,112,123. The cube has three classes of symmetry, which can be represented by coloring the faces. The highest octahedral symmetry Oh has all the faces the same color, the dihedral symmetry D4h comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D2h is also a symmetry, with sides alternating colors. Each symmetry form has a different Wythoff symbol, a cube has eleven nets, that is, there are eleven ways to flatten a hollow cube by cutting seven edges. To color the cube so that no two adjacent faces have the color, one would need at least three colors

6.
Cubic crystal system
–
In crystallography, the cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals, there are three main varieties of these crystals, Primitive cubic Body-centered cubic, Face-centered cubic Each is subdivided into other variants listed below. Note that although the cell in these crystals is conventionally taken to be a cube. This is related to the fact that in most cubic crystal systems, a classic isometric crystal has square or pentagonal faces. The three Bravais lattices in the crystal system are, The primitive cubic system consists of one lattice point on each corner of the cube. Each atom at a point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom. The body-centered cubic system has one point in the center of the unit cell in addition to the eight corner points. It has a net total of 2 lattice points per unit cell, Each sphere in a cF lattice has coordination number 12. The face-centered cubic system is related to the hexagonal close packed system. The plane of a cubic system is a hexagonal grid. Attempting to create a C-centered cubic crystal system would result in a simple tetragonal Bravais lattice, there are a total 36 cubic space groups. Other terms for hexoctahedral are, normal class, holohedral, ditesseral central class, a simple cubic unit cell has a single cubic void in the center. Additionally, there are 24 tetrahedral voids located in a square spacing around each octahedral void and these tetrahedral voids are not local maxima and are not technically voids, but they do occasionally appear in multi-atom unit cells. A face-centered cubic unit cell has eight tetrahedral voids located midway between each corner and the center of the cell, for a total of eight net tetrahedral voids. One important characteristic of a structure is its atomic packing factor. This is calculated by assuming all the atoms are identical spheres. The atomic packing factor is the proportion of space filled by these spheres, assuming one atom per lattice point, in a primitive cubic lattice with cube side length a, the sphere radius would be a⁄2 and the atomic packing factor turns out to be about 0.524. Similarly, in a bcc lattice, the atomic packing factor is 0.680, as a rule, since atoms in a solid attract each other, the more tightly packed arrangements of atoms tend to be more common

7.
Cyclic symmetry in three dimensions
–
In three dimensional geometry, there are four infinite series of point groups in three dimensions with n-fold rotational or reflectional symmetry about one axis does not change the object. They are the symmetry groups on a cone. For n = ∞ they correspond to four frieze groups, the terms horizontal and vertical imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are Coxeter notation in brackets, and, in parentheses and it has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis. Cnv, of order 2n - pyramidal symmetry or full acro-n-gonal group, for n=1 we have again Cs. This is the group for a regular n-sided pyramid. Thus, like Dnd, it contains a number of improper rotations without containing the corresponding rotations. for n=1 we have S2, also denoted by Ci, this is inversion symmetry. C2h, and C2v, of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group, C2v applies e. g. for a rectangular tile with its top side different from its bottom side. In the limit these four groups represent Euclidean plane frieze groups as C∞, C∞h, C∞v, portions of the infinite plane can also be cut and connected into an infinite cylinder. Dihedral symmetry in three dimensions Sands, Donald E, mineola, New York, Dover Publications, Inc. p.165. Coxeter, edited by F. Arthur Sherk, Peter McMullen, thompson, Asia Ivic Weiss, Wiley-Interscience Publication,1995, ISBN 978-0-471-01003-6 N. W. Johnson, Geometries and Transformations, Chapter 11, Finite symmetry groups

8.
Crystal system
–
In crystallography, the terms crystal system, crystal family and lattice system each refer to one of several classes of space groups, lattices, point groups or crystals. Informally, two crystals are in the crystal system if they have similar symmetries, though there are many exceptions to this. Space groups and crystals are divided into seven crystal systems according to their point groups, five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, a lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems, triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, in a crystal system, a set of point groups and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the crystal system. In total there are seven crystal systems, triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, a crystal family is determined by lattices and point groups. It is formed by combining crystal systems which have space groups assigned to a lattice system. In three dimensions, the families and systems are identical, except the hexagonal and trigonal crystal systems. In total there are six families, triclinic, monoclinic, orthorhombic, tetragonal, hexagonal. Spaces with less than three dimensions have the number of crystal systems, crystal families and lattice systems. In one-dimensional space, there is one crystal system, in 2D space, there are four crystal systems, oblique, rectangular, square and hexagonal. The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the table, Note. To avoid confusion of terminology, the term trigonal lattice is not used, if the original structure and inverted structure are identical, then the structure is centrosymmetric. Still, even for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure and this is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the structure, then the structure is chiral

9.
Point groups in three dimensions
–
In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, O itself is a subgroup of the Euclidean group E of all isometries. Symmetry groups of objects are isometry groups, accordingly, analysis of isometry groups is analysis of possible symmetries. All isometries of a bounded 3D object have one or more fixed points. We choose the origin as one of them, the rotation group of an object is equal to its full symmetry group if and only if the object is chiral. Finite Coxeter groups are a set of point groups generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group has n mirrors and is represented by a Coxeter–Dynkin diagram, Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. SO is a subgroup of E+, which consists of direct isometries, i. e. isometries preserving orientation, it contains those that leave the origin fixed. O is the product of SO and the group generated by inversion. An example would be C4 for H and S4 for M, Thus M is obtained from H by inverting the isometries in H ∖ L. This is clarifying when categorizing isometry groups, see below, in 2D the cyclic group of k-fold rotations Ck is for every positive integer k a normal subgroup of O and SO. Accordingly, in 3D, for every axis the cyclic group of rotations about that axis is a normal subgroup of the group of all rotations about that axis. e. See also the similar overview including translations, when comparing the symmetry type of two objects, the origin is chosen for each separately, i. e. they need not have the same center. Moreover, two objects are considered to be of the symmetry type if their symmetry groups are conjugate subgroups of O. The conjugacy definition would allow a mirror image of the structure, but this is not needed. For example, if a symmetry group contains a 3-fold axis of rotation, there are many infinite isometry groups, for example, the cyclic group generated by a rotation by an irrational number of turns about an axis. We may create non-cyclical abelian groups by adding more rotations around the same axis, there are also non-abelian groups generated by rotations around different axes. They will be infinite unless the rotations are specially chosen, all the infinite groups mentioned so far are not closed as topological subgroups of O

10.
Polyhedral group
–
In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. There are three groups, The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron. The reflection symmetries have 6,9, and 15 mirrors respectively, the octahedral symmetry, can be seen as the union of 6 tetrahedral symmetry mirrors, and 3 mirrors of dihedral symmetry Dih2. Pyritohedral symmetry is another doubling of tetrahedral symmetry, S. M. Regular Polytopes, 3rd ed

11.
Cycle graph (algebra)
–
In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle is the set of powers of a group element a, where an. The element a is said to generate the cycle, in a finite group, some non-zero power of a must be the group identity, e, the lowest such power is the order of the cycle, the number of distinct elements in it. Cycles can overlap, or they can have no element in common, the cycle graph displays each interesting cycle as a polygon. If a generates a cycle of order 6, then a6 = e, then the set of powers of a2, is a cycle, but this is really no new information. Similarly, a5 generates the same cycle as a itself, so, only the primitive cycles need be considered, namely those that are not subsets of another cycle. Each of these is generated by some primitive element, a, take one point for each element of the original group. For each primitive element, connect e to a, a to a2, an−1 to an, etc. until e is reached. The result is the cycle graph, when a2 = e, a has order 2, and is connected to e by two edges. Except when the intent is to emphasize the two edges of the cycle, it is drawn as a single line between the two elements. As an example of a cycle graph, consider the dihedral group Dih4. The multiplication table for this group is shown on the left, notice the cycle e, a, a2, a3. It can be seen from the table that successive powers of a behave this way. In other words,2 = a2,3 = a and this behavior is true for any cycle in any group – a cycle may be traversed in either direction. Cycles that contain a number of elements implicitly have cycles that are not shown in the graph. For the group Dih4 above, we want to draw a line between a2 and e since 2 = e, but since a2 is part of a larger cycle. There can be ambiguity when two cycles share an element that is not the identity element, consider for example, the simple quaternion group, whose cycle graph is shown on the right. Each of the elements in the row when multiplied by itself gives −1

12.
Hyperoctahedral group
–
In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930, groups of this type are identified by a parameter n, the dimension of the hypercube. As a Coxeter group it is of type Bn = Cn, as a wreath product it is S2 ≀ S n where S n is the symmetric group of degree n. As a permutation group, the group is the symmetric group of permutations π either of the set or of the set such that π = −π for all i. As a matrix group, it can be described as the group of n×n orthogonal matrices whose entries are all integers, the representation theory of the hyperoctahedral group was described by according to. In three dimensions, the group is known as O×S2 where O≅S4 is the octahedral group. Geometric figures in three dimensions with this group are said to have octahedral symmetry, named after the regular octahedron. In 4-dimensions it is called a symmetry, after the regular 16-cell. In two dimensions, the group structure is the abstract dihedral group of order eight, describing the symmetry of a square. Multiplying these together yields a third map C n →, the kernel of the first map is the Coxeter group D n. In the other direction, the center is the subgroup of scalar matrices, geometrically, in dimension 2 these groups completely describe the hyperoctahedral group, which is the dihedral group Dih4 of order 8, and is an extension 2. V. In general, passing to the subquotient is the group of the projective demihypercube. The hyperoctahedral subgroup, Dn by dimension, The chiral hyper-octahedral symmetry, is the direct subgroup, another notable index 2 subgroup can be called hyper-pyritohedral symmetry, by dimension, These groups have n orthogonal mirrors in n-dimensions. The group homology of the group is similar to that of the symmetric group. This is easily seen directly, the −1 elements are order 2, and all conjugate, as are the transpositions in S n, and these are two separate classes. These elements generate the group, so the only non-trivial abelianizations are to 2-groups, the maps are explicitly given as the product of the signs of all the elements, and the sign of the permutation. Multiplying these together yields a third map, and together with the trivial map these form the 4-group. The second homology groups, known classically as the Schur multipliers, were computed in and they are, H2 = {0 n =0,1 Z /2 n =22 n =33 n ≥4

In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which …

A tetrahedron is invariant under 12 distinct rotations, reflections excluded. These are illustrated here in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through the positions. The 12 rotations form the rotation (symmetry) group of the figure.